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Book: FieldWithOneElement

See: Math, Binomial theorem, Finite fields

Field with one element, {$F_1$}

The field with one element, {$F_1$}, is a nonexistent mathematical concept (fields are supposed to contain at least two distinct elements, 0 and 1) which has spurred quite a bit of research. It suggests itself in different situations as a limiting initial case.

My impression is that it relates to my concept of a God who goes beyond himself into himself, who asks, Is God necessary? Would I be even if I wasn't?


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Readings

Introductory

Overview

General theories

Special aspects

Interpretations of mathematical structures in terms of {$F_1$}.

F1-believers base their f-unny intuition on the following two mantras :

What form does the binomial theorem take in a noncommutative ring? In general one can say nothing interesting, but certain special cases work out elegantly. One of the nicest, due to Schutzenberger [18], deals with variables x, y, and q such that q commutes with x and y, and yx = qxy.

Of course, there is no field F1 with only one element, but there is a trivial ring, and it is merely a convention that we do not call it a field. However, it is an excellent convention, because the trivial ring has no nontrivial modules (if x is an element of a module, then x = 1x = 0x = 0). Calling it a field would not help solve Puzzle 1, since F n 1 does not depend on n. I know of no direct solution to this puzzle, nor of any way to make sense of vector spaces over F1. Nevertheless, the puzzle can be solved by an indirect route: it becomes much easier to understand when it is reformulated in terms of projective geometry. That may not be surprising, if one keeps in mind that many topics, such as intersection theory, become simpler when one moves to projective geometry. (The papers [11] and [22] also shed light on this puzzle by indirect routes, but not by using projective geometry.) Cohn, page 489.

Ideas

Cognitive ideas regarding {$F_1$}.

Finite fields


Dear Harvey,

Thank you for your invitations in your letter below and also earlier, "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"." http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html

You mentioned, in my words, that you are looking for an issue that working mathematicians are grappling with where the classical ZFC foundations are not satisfactory or sufficient. Would the "field with one element" be such issue for you? https://ncatlab.org/nlab/show/field+with+one+element

Jacques Tits raised this issue in 1957 and it has yet to be resolved despite substantial interest, conferences, and long papers. Would that count as a "problem" for the Foundations of Mathematics? It seems that in the history of math it is very easy to simply say "that is not real math" as was the case with the rational numbers, imaginary numbers, infinitesimals, infinite series, etc.

The issue is that there are many instances where a combinatorial interpretation makes sense in terms of a finite field Fq of characteristic q, which is all the more insightful when q=1. For example, the Gaussian binomial coefficients can be interpreted as counting the number of k-dimensional subspaces of an n-dimensional vector space over a finite field Fq. When q=1, then we get the usual binomial coefficients which count the subsets of size k of a set of size n. So this suggests an important way of thinking about sets. However, F1 would be a field with one element, which means that 0=1. But if 0 and 1 are not distinct, then none of the usual properties of a field make sense. Nobody has figured out a convincing interpretation for F1. And yet the concept seems to be pervasive, meaningful and fruitful.

If there was an alternate foundations of mathematics which yielded a helpful, meaningful, fruitful interpretation of F1, would that count in its favor? And if it could do everything that FOM can do, then might it be preferable, at least for some? But especially if that interpretation was shown not to make sense in other FOMs?

I am curious what you and others think.

Andrius

Parsiųstas iš http://www.ms.lt/sodas/Book/FieldWithOneElement
Puslapis paskutinį kartą pakeistas 2019 birželio 30 d., 17:38